Wednesday, 22 January 2014

summation - Power Sum of Integers and Relationship with Sum of Squares and Sum of Cubes

Let σm=nr=1rm.



Refer to the tabulation of the power sum of integers here.




It is interesting to note that



σ1 =12n(n+1)σ2 =16n(n+1)(2n+1)σ3 =14n2(n+1)2=σ12σ4 =130n(n+1)(2n+1)(3n2+3n1)=15σ2 (3n2+3n1)σ5 =112n2(n+1)2(2n2+2n1)=13σ3 (2n2+2n1)σ6 =142n(n+1)(2n+1)(3n4+6n33n+1)=17σ2 (3n4+6n33n+1)σ7 =124n2(n+1)2()=16σ3 ()σ8 =190n(n+1)(2n+1)()=115σ2 ()σ9 =120n2(n+1)2(n2+n1)()=15σ3 (n2+n1)()σ10=166n(n+1)(2n+1)(n2+n1)()=111σ2 (n2+n1)()
i.e.




  • the sum of squares, σ2, is a factor of sum of even powers greater than 2, and

  • the sum of cubes, σ3, is a factor of sum of odd powers greater than 3.





Is there a simple explanation for this, if possible without using Faulhaber's formula and Bernoulli numbers, etc?




and also,




Why does this occur only for σ2,σ3 but not for σ4,σ5, etc?


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